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188 VIII DIFFERENTIAL CALCULUS

13. DIFFERENTIAL OPERATORS

Let A be an open set in R" (resp. C¹), F a real (resp. complex) Banach
space; we denote by 4^P)(A) the set of all/? times continuously differentiable
mappings of A into F. It is clear by (8.12.10) that 4^P) (A) is a real (resp.
complex) vector space; and, more generally, (8.12.10) shows that &#\A)
(resp. 4^P)(A)) is a ring, and $⁽/\A) a module over that ring. For any system

(a_1? . . . , a_n) = a of integers ^0 with |a| = £ a, < p, let M_a = X^¹ X? • • • XJ»
and define D^a or D_Ma as the mapping D^¹ D? • • • D*" of 4^P)(A) into 4^P~^lal)(A).

A linear differential operator is a linear combination D = £ #_aD^a where

t = i
|a| <_/? and the <s_a are continuous scalar functions defined in A; if a_a = 0

for |a| > k and each a_a is (;? — k) times continuously differentiable, D maps
4^P)(A) linearly into 4^P~^fe)(A).

(8.13.1) If the operator £ <2_aD^a is identically 0, f/ze« eac/z of the functions a_af,y identically 0 ?« A. ^a

Write D/= 0 for /(jc) = c • expC^^ + • • • + A_B^), where c ^ 0 is in F
and the l_t are arbitrary constants; we obtain (by Section 8.8 and (8.4.1))

+ • • • + A_n O = 0
identically in A, which is equivalent to £ ^(xyMJ^^ . . . , l_n) = 0; for any

particular x e A, this implies a^x) = 0 for each a, since the A_f are arbitrary.

The coefficients #_a of a linear differential operator are thus uniquely
determined; the highest value of |a| such that #_a ^ 0 is called the order of D.

To each polynomial P = £ £_aM_a of degree ^^ with constant coefficients

we can thus associate a linear operator D_P = £*<*D^a of order ^p; it is

clear that D_Pl+P2 = D_Pl + D_P2, and it follows from (8.12.3) that if P₁P₂had a total degree ^p, then D_PlP2 = D_PlD_P2. In particular, from (8.12.7)
if follows that for fixed £y, the operator /-> D/, where

can be written



	188 VIII DIFFERENTIAL CALCULUS

	13. DIFFERENTIAL OPERATORS Let A be an open set in R" (resp. C¹), F a real (resp. complex) Banach space; we denote by 4^P)(A) the set of all/? times continuously differentiable mappings of A into F. It is clear by (8.12.10) that 4^P) (A) is a real (resp. complex) vector space; and, more generally, (8.12.10) shows that &#\A) (resp. 4^P)(A)) is a ring, and $⁽/\A) a module over that ring. For any system (a_1? . . . , a_n) = a of integers ^0 with \|a\| = £ a, < p, let M_a = X^¹ X? • • • XJ» and define D^a or D_Ma as the mapping D^¹ D? • • • D" of 4^P)(A) into 4^P~^lal)(A). n A linear differential operator* is a linear combination D = £ #_aD^a where t = i \|a\| <_/? and the <s_a are continuous scalar functions defined in A; if a_a = 0 for \|a\| > k and each a_a is (;? — k) times continuously differentiable, D maps 4^P)(A) linearly into 4^P~^fe)(A). (8.13.1) If the operator £ <2_aD^a is identically 0, f/ze« eac/z of the functions a_af,y identically 0 ?« A. ^a

	Write D/= 0 for /(jc) = c • expC^^ + • • • + A_B^), where c ^ 0 is in F and the l_t are arbitrary constants; we obtain (by Section 8.8 and (8.4.1)) + • • • + A_n O = 0 identically in A, which is equivalent to £ ^(xyMJ^^ . . . , l_n) = 0; for any at particular x e A, this implies a^x) = 0 for each a, since the A_f are arbitrary. The coefficients #_a of a linear differential operator are thus uniquely determined; the highest value of \|a\| such that #_a ^ 0 is called the order of D. To each polynomial P = £ £_aM_a of degree ^^ with constant coefficients a we can thus associate a linear operator D_P = £<D^a of order ^p; it is at clear that D_Pl+P2 = D_Pl + D_P2, and it follows from (8.12.3) that if P₁P₂had a total degree ^p, then D_PlP2 = D_PlD_P2. In particular, from (8.12.7) if follows that for fixed £y, the operator /-> D/, where

	can be written